Differentiation

The derivative of $f(\sin x)$ with respect to $g(\sec x)$ at $x = \dfrac{\pi}{4}$, given that $f'\left(\dfrac{1}{\sqrt{2}}\right) = 3$ and $g'(\sqrt{2}) = 1$ is.....

If $y = (x^2 + 1)^{\sin x}$ for $x > 0$ such that $\dfrac{dy}{dx} = y\left[\dfrac{2x \sin x}{g(x)} + \cos x \cdot \log[g(x)]\right]$, then the function $\dfrac{1}{g(x)}$ is...

If $\sqrt{y + x} + \sqrt{y - x} = c$ and $\dfrac{dy}{dx} = K - \sqrt{\dfrac{y^2}{x^2} - 1}$ , then the value of $K$ is

If $\sec^{-1}\left(\dfrac{x^2 + y^2}{x^2 - y^2}\right) = 2a$ such that $y\dfrac{dy}{dx} = x \cdot f(a)$ then the value of $f\left(\dfrac{2\pi}{3}\right)$ is

If $x = a\sin^3 t$ and $y = a\cos^3 t$, then the value of $\dfrac{d^2y}{dx^2}$ at $t = \dfrac{\pi}{3}$ is equal to

If $x = 4t^3 + 3, y = 3t^4 + 4$ and $\dfrac{\frac{d^2x}{dy^2}}{\left(\frac{dx}{dy}\right)^n}$ is constant then the value of $n$ is

If $\sqrt{\dfrac{x}{y}} + \sqrt{\dfrac{y}{x}} = 6$, then $\dfrac{dy}{dx} =$

If $y = \cos^2\left[\cot^{-1}\left(\sqrt{\dfrac{1 - x}{1 + x}}\right)\right]$ then $\dfrac{dy}{dx} = $ ........

The derivative of $\log_{10} x$ with respect to $\log_x 10$ is

If $y = \cot^{-1}\left(\dfrac{1 + \sin 5x}{\cos 5x}\right)$, then the value of $\dfrac{dy}{dx}$ is

Let $x = at^2 - 1$, where $a > 0$ and $y = t^3 + 1$. If at $t = 1$, $\dfrac{d^2y}{dx^2} = \dfrac{3}{16}$, then the value of $a$ is...

If $y = \sin(2\sin^{-1} x)$ then $\dfrac{dy}{dx} =$..

If $3y^2 - 2xy - x = 0$, then the value of $\dfrac{dy}{dx}$ at $y = 2$ is...

If $f(x)$ and $g(x)$ are inverse functions of each other and $f(x) = x + e^x$, then $g'(x) = $...

If $y = \sin^{-1}\left(\dfrac{25 - x^2}{25 + x^2}\right)$, then $y'(1)$ is...

If $x^m + y^m = k, (m \neq 1)$ and $y'' = \dfrac{ax^b}{y^c}$ such that $a + b + c = 0$, then the value of k is...

If $g(x) = (x^2 + 2x + 1)\cdot f(x)$ such that $f(0) = 5$ and $\lim\limits_{x \to 0}\dfrac{f(x)-5}{x} = 4$ then $g'(0) = $

If $x\,e^{xy} = y + \sin^2 x$, then the value of $\dfrac{dy}{dx}$ at $x = 0$ is equal to

Let $t \in (0, 1)$ and $\alpha \in \left(0, \dfrac{\pi}{4}\right)$. If $x = \text{cosec}^{-1}\left(\dfrac{1+t^2}{2t}\right)$, $y = \cot^{-1}\left(\dfrac{\sqrt{1-t^2}}{t}\right)$ and $\dfrac{dy}{dx} = f(t)$, then the value of $f(\tan\alpha)$...

If $f : R \to R$ is an even function then

If $f'(x) = \sin^2 x$ and $y = f\left(\dfrac{2x-1}{x^2+1}\right)$, then $\dfrac{dy}{dx}$ at $x = 1$ is

Let $y = \sqrt[p]{x^3 y}$. If $\dfrac{dy}{dx} = \dfrac{3}{2}$ when $y = 1$, then the value of $p$ is equal to $\ldots$

For $x > 0$ and $(x \log x) < 1$, if $y = \cot^{-1}\left(\dfrac{x - \log x^{x^2}}{\log e^{x^2} + \log x^x}\right)$, then $\dfrac{dy}{dx} = \ldots$

If $f(x) = \cos x \cos 2x \cos 4x \cos 8x \cos 16x$, then $f'\left(\dfrac{\pi}{4}\right)$ is equal to

If $y = [(x+1)(2x+1)(3x+1)\ldots\ldots(nx+1)]^4$, where $n \in N$ and $\dfrac{dy}{dx}$ at $x = 0$ is $2k$, then the value of $k$ is

If $y = x \cdot 7^x$, then the value of $\dfrac{dx}{dy}$ when $x = 1$ is

If $f(x) = \cot^{-1}\left(\dfrac{3x - x^3}{1 - 3x^2}\right)$ and $g(x) = \cos^{-1}\left(\dfrac{1 - x^2}{1 + x^2}\right)$, then $\lim\limits_{x \to a}\dfrac{f(x) - f(a)}{g(x) - g(a)}$, $\left(0 < a < \dfrac{1}{2}\right)$ is

If $y = \sqrt{\cos x^2 + \sqrt{\cos x^2 + \sqrt{\cos x^2 + \ldots \infty}}}$ and $\dfrac{dy}{dx} = \dfrac{f(x)}{2y - 1}$ then, $\int f(x)\, dx = \ldots$

The derivative of $\tan^{-1}\left(\dfrac{\sqrt{1 + x^2} - 1}{x}\right)$ with respect to $\tan^{-1}\left(\dfrac{x}{\sqrt{1 - x^2}}\right)$ at $x = \dfrac{1}{2}$ is

If $y = \dfrac{1}{3x + 5}$, then the value of $\dfrac{d^9 y}{dx^9}$ is..

If $f(x) = \cos x\,\cos 2x\,\cos 4x\,\cos 8x\,\cos 16x$, then $f'\left(\dfrac{\pi}{4}\right) =$

If $y^{\frac{1}{m}} + y^{\frac{-1}{m}} = 2x$ , then $(x^2 - 1)y_1^{\ 2} =$

The derivative of $\log_8(\log_5 x)$ w. r. t. $x$ is

If $y = \tan^{-1}\left[\dfrac{x - \sqrt{1 - x^2}}{x + \sqrt{1 - x^2}}\right]$ , then $\dfrac{dy}{dx} =$

If $y = \cos^{-1}\left(\dfrac{1 - 4^x}{1 + 4^x}\right)$, then $\dfrac{dy}{dx}$ at $x = 1$ is

If $\cot[f(x)] = \dfrac{3x - x^3}{1 - 3x^2}$ and $\sin[g(x)] = \dfrac{1 - x^2}{1 + x^2}$, then $\lim\limits_{x \to t}\dfrac{f(x) - f(t)}{g(x) - g(t)} = \ldots$

Let f(x) be a twice differentiable function such that $f''(x) = -f(x)$, $f'(x) = g(x)$ and $h(x) = \{f(x)\}^2 + \{g(x)\}^2$. If $h(5) = 11$, then $h(10)$ is equal to ...

Let $f(x) = 3x^3 + 4e^{\frac{x}{2}}$ and $g(x) = f^{-1}(x)$. Then the value of $g'(4)$ is equal to...

If $y = \left(\dfrac{ax+b}{cx+d}\right)$, then $2\dfrac{dy}{dx} \cdot \dfrac{d^3 y}{dx^3}$ is equal to

If $y = \cos^{-1}(\sin x)$, where $\dfrac{\pi}{2} < x < \pi$, then $\dfrac{dy}{dx} = ...$

If $y = \log x^x$, then the value of $\left(\dfrac{dy}{dx}\right)^2$ at $x = e^2$ is

If $e^y + xy = e$, then the ordered pair $\left(\dfrac{\text{d}y}{\text{d}x}, \dfrac{\text{d}^2 y}{\text{d}x^2}\right)$ at $x = 0$ is equal to

The derivative of $\sin\left(\log\left(\dfrac{x+3}{x}\right)\right)$ is

If $y = \tan^{-1}\sqrt{\dfrac{1 + \sin 2x}{1 - \sin 2x}}$, then $\dfrac{\text{d}y}{\text{d}x}$ at $x = \dfrac{\pi}{6}$ is

Let $f(x) = x - 5$.If $g(x) = [f(4h(x) + 3)]^2$ and $h(1) = 4, h'(1) = -2$, then $g'(1) =$ .....

If $y = x + e^x$, then $\dfrac{d^2 x}{dy^2}$ is equal to

If $y = \tan^{-1}\left\{\dfrac{\sqrt{1 + x^2} - \sqrt{1 - x^2}}{\sqrt{1 + x^2} + \sqrt{1 - x^2}}\right\}$, where $|x| < 1$, then $\dfrac{dy}{dx}$ is equal to

If $\sqrt{y+x} + \sqrt{y-x} = c$ such that $\dfrac{dy}{dx} = f(x) - \sqrt{[f(x)]^2 - 1}$, then $f(x) =$ ____

If $f(x) = \sqrt{x^2+1}, g(x) = \dfrac{x+1}{x^2+1}, h(x) = 2x - 3$, then $f'\left(h'(g'(x))\right) =$

$\dfrac{d}{dx}\left[\sin^2\left\{\cot^{-1}\sqrt{\dfrac{1-x}{1+x}}\right\}\right] =$ ...